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Verify Lagranges mean value theorem f...

Verify Lagranges mean value theorem for function `f(x)=(x-1)(x-2)(x-3)` on `[0,\ 4]` and find a point `' c '` in the indicated interval:

Text Solution

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Given, `f(x)=(x-1)(x-2)(x-3)` is a polynomial function which is continuous and differentiable everywhere and in `[0,4]`

`f(0)=(0-1)(0-2)(0-3)=-6`

`f(4)=(4-1)(4-2)(4-3)=6`

All condition of Langrage's theorem is satisfied and so there exist `c in (0,4)` such that `f'(c)=(f(b)-f(a))/(b-a)`

`f(x)=(x-1)(x-2)(x-3)`

`f'(x)=(x-2)(x-3)+(x-1)(x-3)+(x-1)(x-2)`

`f'(c)=3c^2-5c-4c-3c+6+3+2`

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